Regularity in $L_p$ Sobolev spaces of solutions to fractional heat equations

Abstract

This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations (*) $Pu+{\partial }_{t}u=f$ on $\Omega \times I$, where $P$ is a nonlocal operator, and $\Omega \subset R^n$, $I\subset R$. 1) For $P$ a strongly elliptic pseudodifferential operator ($\psi$do) on $R^n$ of order $d\in R_{+}$, a symbol calculus on $R^{n+1}$ is introduced, that allows showing optimal regularity of solutions in the scale of anisotropic Bessel-potential spaces $H^{(s,s/d)}$, globally over $R^n\times R$, and locally over $\Omega \times I$, for $s\in R$, $1<p<\infty$. Similar results hold in anisotropic Besov spaces $B^{(s,s/d)}$. 2) Let $\Omega$ be smooth bounded, and let $P$ equal $(-\Delta {)}^a$ ($0<a<1$), or its generalizations to singular integral operators with regular kernels, that are infinitesimal generators of stable Lévy processes. With the Dirichlet condition $u=0$ on $R^n\setminus \Omega$, the initial condition $u{|}_{t=0}=0$, and $f\in L_p(\Omega \times I)$, (*) has a unique solution $u\in L_p(I,H_p^{a(2a)}(\bar{\Omega }))$ with ${\partial }_{t}u\in L_p(\Omega \times I)$. Here $H_p^{a(2a)}(\bar{\Omega })$ equals ${\dot{H}}_p^{2a}(\bar{\Omega })$ if $a<1/p$, and is contained in ${\dot{H}}_p^{2a-\epsilon }(\bar{\Omega })$ if $a=1/p$, but contains nontrivial elements from $d^a{\bar{H}}_p^a(\Omega )$ if $a>1/p$ (where $d(x)=\mathrm{dist}(x,\partial \Omega )$). The interior regularity of $u$ is lifted when $f$ is more smooth.

Minor formulational corrections. To appear in Journal of Functional Analysis

References in corpus (2)

• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Local regularity for parabolic nonlocal operators